Usually, a "damage roll" is considered as "all the dice rolls and damage bonuses that the attack describes". In a simple case, you'd simply do the 1 + 3d4 first and then add the proficiency bonus that corresponds to your level. At first level, that's (1 + 3d4) + 2 .
However, in case of resistances, things get a little complicated. If the target is resistant to poison but not to piercing, how much damage does it take?
The PHB doesn't say anything on the topic. The only similar thing I've found is for things like "3d4 piercing and poison damage", in which case the target would have to have resistances for both effects for it to apply. Extending that logic, one could argue that you apply resistances for each damage type individually. But where does the proficiency bonus come into effect then?
I would say the issue is ambiguous enough that the final decision should fall with the DM and be maintained as a house rule. I would suggest the following rule based on my own reasoning:
The Ranger's proficiency bonus applies to the skill-based aspect of the beast's attack.
Effect: For this example, the snake's ability to bite is enhanced by the Ranger's ability to command it, therefore the piercing potion (the bite) is enhanced by the proficiency bonus while the poison damage (the snake's venom) is dealt as-is. For someone resistant to poison it means the target would take half of the 3d4 poison damage, but the full (1+proficiency) piercing damage.
Rationale: The Ranger's understanding of its snake companion and his ability to train and command it has made the companion more skillful with its attacks. However, the snake's body still produces the same venom as it would if the snake was just a wild animal. Therefore the bonus to the snake's damage should apply to the physical portion unless stated otherwise.
To further illustrate the point, let's consider a Fighter with a flaming sword that deals 2d6 slashing + 1d8 fire damage. When attacking he adds his Strength modifier to the damage. The Fighter's ability to wield a sword is what causes the Strength bonus, so the extra damage would naturally be applied to the slashing part. If a target is resistant to fire, it only resists the 1d8. The snake is similar, with its fangs as its weapon, the poison as the extra effect, and the Ranger's proficiency bonus as the added skill damage.
Counter-rationale:
One could argue that a beast with a purely non-physical attack (e.g. breathing fire or spitting poison) should not receive the bonus. I would suggest that perhaps the Ranger trained the beast into using its natural resource more skillfully (e.g. aiming for the eyes) and that's why the bonus applies (as the original RAW rule intended).
One could also argue that if a damage roll has two types of physical attack that the bonus should apply to both. That would conflict with the RAW rule of "apply the bonus once per damage roll". I would probably assign the bonus to the first type. I haven't ever even heard of such an attack, though, and having a beast with such an attack AND an enemy with a resistance to one of the two is a situation that I'm sure will not arise often.
Multiple attacks:
As the RAW rule intended, this effect is applied once per the beast's attack. That means that, although the snake's attack contains two damage types, it is still a single attack and gains the proficiency bonus once. If a beast can make multiple attacks as its action, though, it gains the bonus once per each attack, but it has to make an attack roll for each attack as well.
Alternative house rule
If the damage roll consists of multiple damage types and the target resists only one of them, apply the proficiency bonus to the non-resisted part of the damage. Only when the target resists all of the damage types that are part of the attack is the proficiency bonus included in the reduced damage calculation.
So, going back to the Flying Snake example, if the target resists poison, it's the same as the former rule: (1+proficiency) + (3d4)/2, rounded down.
If the target resists piercing instead, the proficiency bonus is now applied to the poison damage: 1/2 + (3d4 + proficiency). Because we round down, that means the piercing part is essentially nullified (which you could role-play as the bite was never felt, turning a resistance into a stealthy advantage).
Only if the target has both piercing and poison resistance is the attack fully resisted, and we go back to the standard ((1 + 3d4) + proficiency)/2, rounded down.
I like the first rule better for role-playing purposes, but this one is quick to resolve and doesn't require any consideration as to what a "skill-based aspect" of the attack is.
It gets the benefits immediately
You get the Companion's Bond feature at 3rd level, at the same time as you gain the Animal Companion feature.
The Companion's Bond feature says:
In addition to the areas where it
normally uses its proficiency bonus, an animal
companion also adds its proficiency bonus to its
AC and to its damage rolls.
This applies from 3rd level and onwards, so it gets a +2 to its attack rolls, ability and skill checks, saves, AC and damage, and loses Multiattack if it had it.
Once you reach 4th level, and every level beyond that, it gets an extra hit die:
For each level you gain after 3rd, your animal
companion gains an additional hit die and
increases its hit points accordingly.
This means that your giant weasel's stats look like this:
Giant Weasel
Medium beast, unaligned
Armor Class 15 (13 +2 proficiency)
Hit Points 9 (2d8)
Speed 40 ft.
Ability Scores
(+2 proficiency to all saves)
STR 11 (+0)
DEX 16 (+3)
CON 10 (+0)
INT 4 (-3)
WIS 12 (+1)
CHA 5 (-3)
Skills Perception +5, Stealth +7
Senses darkvision 60 ft., passive Perception 13
Languages —
Challenge 1/8 (25 XP)
Keen Hearing and Smell: The weasel has advantage on Wisdom (Perception) checks that rely on hearing or smell.
Actions
Bite: Melee Weapon Attack: +5 to hit (+3 +2 proficiency), reach 5 ft., one target. Hit: 7 (1d4 + 3 + 2 proficiency) piercing damage.
Best Answer
Yes
It is important to note that it says, without condition, that the companion uses the Ranger's proficiency bonus. This means that, from this point on, it's proficiency bonus is the Ranger's proficiency bonus. Think of it as erasing the value on the creature sheet, replacing it with the Ranger's value, and then recalculating everything from there.
Upon looking at the UA document again, this is further supported as such:
While the wolf has a listed proficiency bonus of +2, nowhere is it stated that this bonus is already included in the damage of the wolf's attack. We may infer that this damage is from the wolf's DEX mod, as is supported by the fact that the base to hit is +4, which would be +2 from proficiency and +2 from DEX, matching expectations for standard application of proficiency bonuses. The wolf's damage should be increased by a flat 3 points, making the roll 2d4+5, the same as you have stated. Furthermore, applying +3 to the base 13 also matches your stated 16 AC, and replacing the +2 with proficiency with +3 also results in a +5 to hit when added to the DEX mod. It is not explicitly stated that the bite is a DEX-based attack, but this is the only value that makes sense from the creature stats. Your math looks right.
Bonus Answer: No, the damage dice of the wolf's bite attack do not scale. It is important to consider that the wolf companion is not intended to be as powerful as an extra PC, but is instead part of the power of the Ranger to whom it belongs. Applying proficiency bonus to a damage source is a benefit other classes do not have, and this comes from an attack that does not require the use of any of the Ranger's actions. Also, consider that this attack has the potential to knock a target prone, leaving it more vulnerable to melee attacks until it has the opportunity to stand. Such a tactical advantage is certainly not weak.